How to Cut Any Straight-Sided Shape out of Paper with a Single Cut

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Here’s a fact that would have blown your 7-year-old self’s mind: you can cut any straight-sided shape out of a piece of paper with a single cut. Just one snip of the scissors and you can create anything from a star to a snowflake to a butterfly.

The technique of folding up a piece of paper and then cutting it to create a shape is called the “fold-and-cut” method.

The “fold-and-cut” problem asks: What shapes can you make with the fold-and-cut method? Humans have been asking this question for a surprisingly long time. The first published reference on record is from a Japanese puzzle book from the 18th century.

In more recent years, the fold-and-cut problem has gone under study by professional mathematicians.

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In 1999, mathematicians Erik Demaine, Martin Demaine, and Anna Lubiw formulated a proof that found that any straight-sided shape can be created through the fold-and-cut method.

That’s right — you can use this method to create literally any shape that you can dream up, as long as it has straight sides rather than round. You can create a swan, an angelfish, and much more by making a single cut in a piece of paper.

Erik Demaine has published printable images that show exactly how to make these shapes. These images can be a little confusing at first glance, so the below video also demonstrates how to make the swan.

There’s also a new solution for the problem called the “disk-packing” approach, which involves placing disks on top of paper.

Time to make the best paper snowflakes that the world has ever seen!